Calculator for computing volumes of revolution



Aug. 30, 1955 M. D. BRAlD CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTIONFiled Oct. 9, 1955 5 Sheets-Sheet l liizranlsir Murray D. Braid M. D.BRAID Aug. 30, 1955 CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION 5Sheets-Sheet 2 IFZFEWZET Murray .2). Braid MMWW M HZLZHE Flled Oct 91953 Aug. 30, 1955 M. D. BRAID 2,716,522

CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION Filed Oct. 9, 1953 5Sheets-Sheet 3 0, 1955 M. D. BRAID 2,716,522

CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION Filed 001;. 9, 1955 5Sheets-Sheet 4 W Q 7 1% w 4 Y EYE 272277 Murray .D. .Bmz'a M MM-M M 9 Wb H H ZZ 175 I Aug. 30, 1955 M. D. BRAID 2,716,522

CALCULATOR FOR COMPUTING VOLUMES OF REVOLUTION 157.273 17ZUT Murray .23.Bra/I0 wva MM W Z:/7 7 HZZL LE United States Patent 0 CALCULATOR FORCOMPUTING VOLUMES OF REVOLUTION Murray D. Braid, Mentor, Ohio, assiguorto Thompson Products, Inc., Cleveland, Ohio, a corporation of OhioApplication October 9, 1953, Serial No. 385,211

10 Claims. (Cl. 235-61) The present invention relates to calculatingmachines, and more particularly relates to calculating machines for usein drafting rooms as an aid in determining volumes of revolution.

In modern industry the necessity of determining intricate volumes ofrevolution, or volumes of revolution of intricate parts, arisingconstantly. For example, in the manufacture of forged metal parts suchas automotive engine valves, valve tappets and other similar metal partshaving a symmetrical configuration, it is desired that the amount ofmetal in the finished part be known. When the finished volume is known,the initial billet or blank from which the part is to be forged may becut to the proper size so that flash and waste material may besubstantially eliminated.

The desirability of the prevention of such waste has long beenunderstood in industry and the relatively tedious methods ofapproximation heretofore utilized in determining volumes, of revolutionhave been followed in the design and drafting rooms in order to permitthe elimination to the greatest possible extent of waste material in themanufacturing processes. While, of course, simple volumes of revolutionsuch as for example those of cylinders, cones and spheres may readily bedetermined without diliiculty, in most instances the configuration ofthe part is made up of curves, angles and other irregularities whichmake it substantially impossible to calculate the volume accurately in areasonable period of time.

An object of the present invention is, therefore, to provide an easilymanipulated apparatus of convenient size which will automaticallydetermine the volume of revolution of any curve that can be drawn onflat paper.

A further object of the invention is to provide an extremely compactcalculating apparatus capable of use on ordinary drafting boards for thedetermination of volumes of revolution.

Still a further object of the present invention is to provide anextremely compact calculator capable of automatically solving theequation for predetermined range of values of x and r and in which r isnon-uniformly variable.

And yet another object of the present invention is to provide a dualintegrating apparatus capable of determining the volumes of revolutionof a predetermined curve by the manual manipulation of a portion of theapparatus to trace the curve.

A further object of the present invention is to provide a calculatingapparatus capable of determining the volume of revolution of any curveby an operator having no knowledge whatever of the mathematics involved.

A further object of the present invention is to provide a calculatingmachine constructed to move at a uniform rate along a carrierpositionable in adjusted relation to 2,716,522 Patented Aug. 30, 1955 acoordinate of a drawn curve and having an indicator adjustable to followthe curve in the direction of the other coordinate whereby tracing thecurve by said pointer will automatically determine the volume ofrevolution of the curve about the first coordinate.

Still other and further objects of the present invention will, ofcourse, become apparent to those skilled in the art from a considerationof the attached sheets of drawings in which a preferred embodiment ofthe present invention is shown by way of illustration only.

On the drawings:

Figure 1, is a perspective view of the calculator constructed accordingto the present invention.

Figure 2, is a plan view of the body of the apparatus, with the coverremoved to disclose the internal structure thereof.

Figure 3, is a front elevation view in detail shown in partial crosssection.

Figure 4, is a sectional view taken along the line IV- IV of Figure 2showing the gearing connections of the first integrator in detail.

Figure 5, is sectional view in detail taken along the line V-V of Figure2.

Figure 6, is a schematic showing of the basic interrelationship betweenthe various components of the present invention.

Figure 7 is a gearing schematic of the apparatus, further illustratingits construction; and

Figure 8 is a diagrammatic drawing of the mechanical integrator unitsutilized in the present invention.

As shown on the drawings:

The problem faced in the present instance, and solved by the computer ofthe present invention, is the value of the volume of revolution of acurve such as the curve 8 in Figure 1, about an axis of revolutiondenoted X. Mathematically, utilizing the symbols: r=radius of aparticular point, or its distance away from the axis of revolution x,2:: distance along the axis x-x, and dr and dx equal small increments inthe respective values of r and x respectively, the volume of revolutionV may mathematically be stated as follows:

V=21rf f rdrdx In solving the above double integral equation the presentinvention contemplates the utilization of two ball and disk typemechanical integrators. Such an integrator is shown in diagrammatic formin Figure 8 and although its operation is generally known in the art,the fundamental theory thereof Will hereinafter be reviewed. Assumingthe shaft 0 is arranged to drive a second cylindrical shaft through agear ratio N, then for a rotation d" of the driving shaft 0, therotation of the driven shaft will be Ndli. When the gear ratio changesduring the driving action of the shaft 0, the total rotation of shaft 1:equals the integral of Ndfi.

In the use of an integrator of the ball and disk type the gear ratio Nequals the distance U of the transmitting ball B from the axis ofrotation of the disc divided by.

R, the radius of the driven shaft cylinder Since the value R is aconstant it will thus be seen that for a rotation of 110 with a changinggear ratio, the total rotation of the cylinder will be 1 fUde Since, inmechanical integrators, it is desirable to pro vide a change in theposition of the ball B by means of a lead screw or a similar threadedapparatus which provides axial movement by rotation, the factor U may beexpressed in terms of rotation of a shaft. Thus U can be said to equalwhere P is the pitch of the screw measured in terms of turns per inch ofdisplacement along a shaftv which is turning in accordance with thevariable of 1ntegration.

Substituting in the equation Ude Utilizing this with an integratingapparatus in actual practice, 1 preferably provide a mechanicalintegrator we therefore have constructed to provide two completerevolutions of the disk 0 per inch of travel or distance r as viewed inFigure 1 thus making r directly proportional to 0. Movement of thepointer P .is thus translated into two revolutions of the disk per inchof r. In the calculator of the present invention, which is solving forthe integral of rdr, r is the 3 variable of integration and is also theinput into the first disk 0. Therefore, for the first integrator vequals 01 which is in turn directly proportional to r. In thisconnection it is noted that hereinafter the subscript 1 indicates thatthe component is for the first integrator of the series and subscript 2indicates that component under consideration applies to the secondintegrator of the series.

In view of the above relationships,

1 4 1 EFT! f l l or, integrating,

Since as was calculated above,

, hi 21mm then 2 wRlPlPzRzl The exact solution of this equation isdependent upon the limitations imposed by the calculator rangerequirements. The limits of the values r and x in the equation V=21rffrdrdx are governed by the size of object to be measured. For the sake ofthe purposes for which the computer of the present invention wasconstructed, it was decided that the maximum value of x=3 inches and themaximum value of r=2 inches were satisfactory. However, in order to makethe computer easier to handle, the gearing components of a computerconstructed according to the present invention were designed to producea reading at the output shaft 92 equal to cubic inches X100 when thepointer traces a layout of the curve which is five times its regularsize. Thus, the calculator mentioned will operate over a range of xequals 15 inches and 1' equals 10 inches maximum. While this calculatorhas proven very satisfactory, it is noted, of course, that the actualmaximum dimensions can be varied in constructing the apparatus.

Since, as was noted above, in the present instance the input to thefirst disk, or 61 is directly proportional to r, or, in other words61:10, and since the values of RiliRzPz are all constant, it istherefore clear that where dti'z is the angular input to the secondintegrator disc and is directly correlated in a constant manner tomovement of the computer in the xx direction as viewed in Figure 1. Onthese bases it is clear that the dual integrators may be arrangedthrough appropriate constant gearing to provide the solution to theintegral, as set out above.

While variations may of course be utilized in constructing the geartrains correlating the integrators with the 1' pointer P and with themovement of the entire mechanism in the xx direction, for purposes ofillustration a sample gear train is hereinafter set forth.

A commercially available integrator of the type above discussed andhaving a ball carriage travel of .75 inch and an output factor of 1 maybe used. Since, for the sake of convenience, both of the integratorsutilize the same structural components, the values of R1 and R2 areequal. Further, the lead screws utilized for modifying the position ofthe ball carriage are also equal and are chosen to be 32 threads perinch. Under these circumstances, the equation will equal, withsubstitutions,

In order to obtain a direct relationship between (162 and x, in terms ofx, it is necessary to apply the computer range limits to the aboveequations. Since, from Q 2R P 20 above, and since the output 1 of thefirst integrator equals rdr 2 then the equation V=21rjfrdrdx Due to thechosen ratio of 2:1 between inches of r and turns of the input shaft 61,r equals its maximum of "10 inches then 61 equals 20 turns. Therefore,at the limit position the equation 2 t e 200 f i z becomes 400 d d0 2d0Since the rotation of the output shaft is to equal the volume ofrevolution V or a multiple thereof, and since the rotation of the outputshaft of the second integrator equals 02, then Applying the limit 0:20,and inserting the value of d2 equals 21102, we have 112 equals2d02=801rdx, or, in other words we find that the rotational input 1102to the disc of the second integrator should bear a relationship of 401rturns per inch of travel in the x direction.

Substituting this value in the previous equation of When 0:20, or themaximum of the assumed sample conditions adopted, then 2=801rdx.

From the above equations it will be apparent that the actual gearingvalues of the various elements in the mechanical integrator may be asfollows, referring to the schematic showing of Figure 6: The input shaft10 l to the first integrator will be geared by a two to 1 ratio to thepointer 11 so that one inch of movement of pointer 11, measuring 1',will cause 2 turns of the shaft 10. Rotation of the shaft 10, whichrotation is termed 0, is fed directly to the input shaft 12 of the disk13 of the first integrator generally indicated at 1. As will be recalledfrom the calculations above the quantity 61 is placed in one to oneratio onto the disk 13 and hence the gearing connection between theshafts 10 and 12 is a one to one direct drive. The quantity v, whichindicates the movement of the ball carriage relative to the disk 01,will be recalled from the above calculations to be controlled by a leadscrew having a pitch equal to 32 turns per inch. This lead screw isshown at 14 and connects the shaft 10 to the integrator 1 to provide 1inch of movement of the integrator 1 for each 32 turns of the shaft 10.

The output shaft 15, the rotation of which has been indicated incalculations above to be will form this mechanical arrangement providean output to a connecting shaft 16. The connecting shaft 16 provides theinput v2 into the second integrator generally indicated at 2 byproviding the variable of integration to the ball carriage through thelead screw 17 again having a pitch of 32 turns per inch.

The quantity 02, or the input into the disk 18 of the integrator 2, aswas found above, bears a relationship with movement along the x axis of401r turns of the shaft 19 per inch of movement of the pointer 11 in thex direction. Thus the shaft 19 is geared to a rack lying in the xdirection to provide the 401r turns per inch and represents the dxquantity which will, in the mechanical integrator, be a uniform rate ofmovement in the x direction. The output from the second integrator, 2,is taken from the cylinder shaft 21 which represents, in thecalculations above, 2. This output equals, according to the developmentabove,

1 2 2 fthdG; OI KfT dx and is applied through suitable gearing indicatedat 22 of Figure 6 to a counter shaft 23 which provides a numericalindication capable of multiplication by a constant to provide the actualvolume of revolution of the curve in its exact dimensions. It is ofcourse to be understood that the counter can be adapted to provide adirect reading requiring no multiplication by a constant, through theappropriate use of gear trains between the counter and the output shaft21.

While of course, various gear trains can be utilized in obtaining thenecessary ratios between the components, the train shown in the Figure 7gearing schematic diagram has proven very satisfactory in the examplecalculator. As may there be seen, the pointer 11 is provided with rackteeth 25 which engage a gear 26 fixedly secured to the shaft 27. Thegear 26 is provided with 20 teeth and the gear 28 secured to theopposite end of the shaft 27 is provided with 40 teeth. Gear 28 in turnis enmeshed with the 14 tooth gear 29. The shaft 30 which carries thegear 29, carries also a gear 31 which drives the shaft 10A through thegear 32 having 32 teeth. The rack or pointer 11 has teeth out on adiametral pitch of 32, as do all of the remaining gears in the abovegear train. Thus, the rack has teeth per inch of its length. On thisbasis it is clear that movement of the rack 1 inch will provide 2 turns.

I in the gearing schematic Figures 6 and 7 wherein the output 1 is fedthrough a screw connection 17 having 32 threads per inch into the ballcarriage of integrator 2. The quantity dx which is a uniform rate ofchange along the axis x in the present example, is provided by means ofa constant speed motor 35. According to the above calculations the motoris constructed to turn the shaft 20 at a rate which is equal to 401:-times the travel in inches along the x dimension.

In mechanically constructing this arrangement it is therefore desirablethat the motor be utilized to provide simultaneously the power to theshaft 20 and also the power to move the entire apparatus in the xdirection. This. may be accomplished through the gearing shown by way ofexample in the gearing schematic Figure 7. There, the motor 35 drives apinion gear 36 having 30 teeth and which is enmeshed with a gear 37mounted on the countershaft 38. The gear 37 is provided with 32 teethand a second gear 39 also secured to the shaft 38 is provided with 44teeth and meshes with the gear 40 on the shaft 20. The gear 40 isprovided with 14 teeth and hence for each revolution of the motor shaftgear 36, the shaft 20 will rotate times.

Parallel gearing is provided for driving the entire carriage in the xdirection along a rack designated 41 and comprises a worm 42 aflixed tothe shaft 38 and in contact with a worm wheel 43. The worm wheel 43 iscaused to rotate once for each 40 revolutions of the worm 42 and thegear 43 is caused to rotate l revolution for each 1 inch travel of therack 41 relative thereto. This one revolution of gear 43 per inch oftravel of the rack 41 is provided by the gear 44 fixedly secured to thegear 43, and idler gear of 45 and a rack gear of 46 having 44 teethwhich in turn is directly engaged with the rack 41 by means of a pinion47 having 42 teeth. The rack is provided with teeth per inch, having adiametral pitch of 32. It will thus be seen that the shaft 20 whichturns at a rate of 1r times the rate of shaft 38 which in turn rotatesat 40 times the rate of rotation of the gear 43 will necessarily rotateat a rate of 401.- revolutions per inch of travel on the rack 41. Theinput motor 35 may, of course, turn at any speed desired since theratios necessary to the proper performance of the apparatus relate onlyto the relationship between the shaft 20 and the rack 41 rather than tothe rotation of the motor shaft.

Arrangement of the gearing shown in the schematic Figure 7 into acompact mechanical apparatus may be clearly seen from a consideration ofFigures 2, 4 and 5. In these views which show the calculator with thecover, 8, removed, the numbers of the various components utilized in theschematic diagrams in the Figures 6 and 7 have been utilized, althoughthe physical arrangement of the component parts is modified to provide avery compact unit.

The carriage board or base upon which the calculating components aremounted is generally indicated at 3 and is movable in the directions ofthe arrow 4 on the carriage rails 5 and 6 by means of the wheels 7. Thepointer P connected to the rack 11 is slidably carried in the slideblocks 9 secured to the housing base 3 and may be manually reciprocatedin the guides as the pointer P is guided over the surface of the curvebeing traced. This reciprocating movement transversely of the rails 5and 6, of the rack 11, constitutes the variable input to the calculatorin the r direction and is fed into the integrator 1 at the shafts 12 and14 which comprise the shaft connected to the integrator disk and theball carriage respectively. The connections between the rack and theshafts 12 and 14 comprise the gearing shown in the schematic Figure 7and are mounted in the actual mechanical construction as follows as maybe seen from Figures 2, 4 and 5.

The gear support 50 carries the shaft 27 having the gear 26 mountedthereon for cooperation with the rack gear teeth 25. At its opposite endthe shaft 27 carries a gear 28 meshed with the gear 29 carried by thestub shaft 30. The stub shaft 30 carries the gear 31 which turns thegear 32 and also turns the gear 32a through the intermediate idler 33.The output of the gear 32a is fed through the shaft b and the gearlng 34to the shaft 12, while the output gear 32 reciprocates shaft 14 by meansof the threaded lead screw 13 having threads which cooperate withinternal threads on the gear. The output shaft of the integrator 1 1sdirectlycoupled to a sleeve nut 16 which is threaded on to the shaft 17of the integrator 2. As explained above, the shaft 17 has 32 threads perinch thereon and is moved axially upon rotation of the shaft 15. Thusthe integral of rdr is fed into integrator 2 as a variable effecting theposition of the ball carriage relative to the disk 18 of the secondintegrator.

The dx component which comprises the constant rate movement of thecarriage base 3 relative to the rack 41 on the rail 6 is fed into thesecond integrator at the shaft 20. This gearing may best be seen from aconsideration of Figures 2 and 5 wherein the gear support 51 carries thegear 39 on the shaft and it further carries aifixed thereto the worm 42meshed with the gear 43. The support 51 also carries the shaft 43 whichsupports the gears 43 and 44 at right angles to the worm 42. The gear 44meshes with the rack 41 through the idler gear 45, mounted on the stubshaft 49, and the gears 46 and 47.

Power input for moving the carriage along the rail 6 is, as was abovenoted, provided electrically in form of the motor 35 which drives theshaft 38 through the gear 36 and 37 after passing through the reductiongear indicated at 35a. The output of the second integrator shaft 21 isfed into the revolution counter 23 by a pair of gears 52 having equalnumbers of teeth. As explained above this relationship provides anumerical figure at the counter equal to the volume to be determinedtimes 100. Of course a to 1 gear ratio could be utilized in place of thegear 52 to provide a direct reading at the counter 23 if so desired.Further, since varying the magnification of the original drawing willchange the volume by a factor not influenced by integration, varyingmagnifications of the original drawings may be used merely by changingthe gearing 52 or my modifying the reading at 23 by an appropriateconstant factor.

In operation, the apparatus is set up as shown in Figure 1. As may therebe seen, the rails 5 and 6 are placed parallel to the x axis, or axis ofrevolution, of the curve S. The pointer P is then positioned at theorigination point 0 of the curve S. The motor 35 may then be energisedand as the base 3 is moved along the rails 5 and 6, the pointer P ismoved along the curve S manually. When the pointer reaches the end ofthe curve S, the reading of the counter 23 is taken. For conveniencesake, a reset knot 23b is provided on the counter to permit setting thecounter at zero at the initiation of tracing.

It has been found that the above apparatus will calculate volumes ofrevolution with an average error of approximately .3 to 1.2% maximumdepending mainly upon the ability of the operator of the machine tofollow the curve exactly with the pointer. Of course in setting theapparatus up to determine the volume of revolution it is essentiallythat the line xx of the drawing, namely the axis of revolution, beexactly parallel with the carriage rails 5 and 6, and that the pointer Pbe positioned on the axis xx at the end of the curve S.

The electric motor 35 and the reduction gearing 35a were, in the exampleapparatus, constructed to provide a rotary input to the gears 36 ofapproximately revolutions per minute. Using the gearing set forth in theexample above, this provides a movement of approximately 2.7 inches perminute of the carriage 3 in the xx directional on the rails 5 and 6.This feed may of course be slowed by providing a motor 35 of a differentspeed or a transmission 35a of a different gearing ratio. It has beenfound however that a movement of 2 to 3 inches per minute is slow enoughfor the average operator to trace the curve S with reasonable accuracy.It is to be noted, of course, that the electric motor 35 may becompletely eliminated and the carriage 3 moved manually if so desired.However, in view of the care with which the pointer P should bemanipulated, it is desired that the operator be required to manipulateit only rather than provide movement in the x direction also.

As an aid in the manipulation of the pointer, I have provided anauxiliary hand wheel 60 which is geared to the rack 25 through the shaft61 and idler gear 33. The hand wheel is supported conveniently on thecarriage 3 by supports 62, 63 secured thereto. By rotational movement ofthe hand wheel, the pointer P may be caused to follow the curve S and,simultaneously, the input will be fed to the integrator 1.

It will thus be apparent that I have provided an extremely efficientcompact, and easily operated calculating apparatus. This will be morefully appreciated when it is understood that the dimensions of thebaseboard 3 of the housing are, in the example above set out, only 7%inches wide and 13 /2 inches in lengh. The rails 5 and 6 may of coursebe any length desired but in view of the fact that the apparatus set outin the example above was constructed to provide a maximum travel of 15inches in the xx direction, it is clear that a length of approximately30 inches is entirely satisfactory for the rails. Under thesecircumstances it is clear that the entire apparatus is substantiallyless than 3 feet in length and hence may readily be utilized with theordinary drafting boards in use today.

Further, the apparatus in the present invention is extremely simple tooperate since it is only necessary that the operator trace the curveproperly in order to provide an extremely accurate reading of the volumeof revolution. Approximately 5 minutes time is all that is required tocompletely trace a curve and hence to calculate the accurate volume ofrevolution. Since the calculation of volumes of revolution havingirregularly curved surfaces by approximation requires several hours timewhen carefully done, and even then the answer arrived at is onlyapproximate, it will be apparent that the use of the calculator of thepresent invention has greatly facilitated the accurate computation ofvolumes of revolution and has hence materially accelerated the designtime in preparing structures for manufacture.

It is of course to be understood that the specific gear trains set outin the above example are exemplary only and may be modified according tospace requirements. Further, it is also clear that modifications couldbe made in the ratio of movement of the pointer relative to the rotationof the input shaft 12 of the integrator 1 and that threads of differentpitch could be utilized at 14 and 17. These changes, which providedifferent constant factors in the equations, would necessarily require amodification of the gearing which accompanies these changes. However,utilizing the formulas set out and the interrelationship between theintegrators above described, it is clear that the volume of revolutionmay accurately be obtained through the mere tracing of the curve S byreciprocating rack 11 and the movement of the carriage 3 in the xxdirection. It is of course, to be further understood that othermodifications and variations may be made within the scope of theconcepts of the present invention.

I claim as my invention:

1. A calculator for the automatic computation of the volume ofrevolution of a curve S about an axis x and having a varying radius 1',which comprises a tracer point for tracing said curve, said tracer beingconnected to the disk and the ball carriage of a ball and diskintegrator to provide a varying output at a rotating output shaft, arail means for carrying said integrator in a direction parallel to the xaxis, means for moving said carriage and said integrator along saidrail, a second ball and disk integrator having its ball carriageconnected to the output shaft of the first integrator and having itsdisk geared to said rail to correlate the rotation of said disk with thetravel of said carriage relative to said rail, and a revolution countersecured to the output shaft of said second integrator for indicating thevolume of revolution of said curve upon completion of the tracingthereof.

2. A calculator for the automatic computation of the volume ofrevolution of a curve about an axis comprising a first mechanicalintegrator having a rotating disc drivingly connected to a rotatingoutput shaft by an adjustably positioned ball, means for tracing saidcurve and positioned for movement perpendicular to said axis, secondmeans connecting said tracing means to said ball for varying itsposition in response to the variation in distance of points in saidcurve from said axis, third means connecting said tracing means to saiddisk for rotating said disc in response to said distance fourth meansconnecting said output shaft to the ball of a second integrator, fifthmeans for driving the disc of said second integrator in direct relationto travel of said tracing means along said axis and counter means drivenby the output shaft of said.

second integrator for recording a value directly proportional to thevolume of revolution of said curve about said axis.

VII

3. Apparatus for solving the integral Kfr dx comprising first and secondball and disc type integrators arranged in series with the output of thefirst providing the variable of integration for the second, and tracingmeans movable along perpendicular axes r and x, first means translatingmovement of said tracing means along the r axis to the ball and to thedisc of the first integrator and second means translating movement alongthe x axis to the disc of the second integrator and counting meanssecured to the output of the second integrator for indicating thesolution.

4. Apparatus for solving the integral comprising first and second balland disc type integrators arranged in series with the output of thefirst providing the variable of integration for the second, and tracingmeans movable along perpendicular axes r and x, first means translatingmovement of said tracing means along the r axis to the ball and to thedisc of the first integrator and second means translating movement alongthe x axis to the disc of the second integrator and counting meanssecured to the output of the second integrator for indicating thesolution, said first means comprising a rack reciprocatable along the raxis and geared directly to the disc of said first integrator to causerotation thereof and geared to the ball of said first integrator tocause reciprocation thereof responsive to movement of the rack.

5. Apparatus for solving the integral comprising first and second balland disc type integrators arranged in series with the output of thefirst providing the variable of integration for the second, and tracingmeans movable along perpendicular axes r and x, first means translatingmovement of said tracing means along the r axis to the ball and to thedisc of the first integrator and second means translating movement alongthe x axis to the disc of the second integrator and counting meanssecured to the output of the second integrator for indicating thesolution, said second means comprising a fixed rack extending parallelto the x axis and gears connecting said rack to the disc of said secondintegrator whereby movement of the tracing means along said x axis willcause rotation of the disc of said second integrator in proportion tothe movement along said x axis.

6. Apparatus for solving the integral comprising first and second balland disc type integrators arranged in series with the output of thefirst providing the variable of integration for the second, and tracingmeans movable along perpendicular axes r and x, first means translatingmovement of said tracing means along the r axis to the ball and to thedisc of the first integrator and second means translating movement alongthe x axis to the disc of the second integrator and counting meanssecured to the output of the second integrator for indicating thesolution, said first means comprising a rack reciprocatable along the raxis and geared directly to the disc of said first integrator to causerotation thereof and geared to the ball of said first integrator tocause reciprocation thereof responsive to movement of the rack, saidsecond means comprising a fixed rack extending parallel to the x axisand gears connecting said rack to the disc of said second integratorwhereby movement of the tracing means along said x axis will causerotation of the disc of said second integrator in proportion to themovement along said x axis.

7. A calculator comprising a platform mounted for reciprocation along afixed guide rail, a first rack mounted on said platform forreciprocation along an axis perpendicular to said rail, a firstintegrator having a rotating disc, an output shaft rotating at rightangles to the axis of the disc and a movable ball positioned forproviding a variable speed drive between said disc and said outputshaft, first gears connecting said first rack to said disc for rotationthereof, second gears connecting said first rack to said ball for movingits position, a second integrator having a second output shaft, a secondmovable ball and a second disc, at second rack secured to said rail,third gears mounted on said platform and connecting said rail to saidsecond disc to rotate said second disc, fourth gears connecting saidfirst output shaft to said second ball and a counter actuated by saidsecond output shaft.

- 8. A calculator comprising a platform mounted for reciprocation alonga fixed guide rail, a first rack mounted on said platform forreciprocation along an axis perpendicular to said rail, a firstintegrator having a rotating disc, an output shaft rotating at rightangles to the axis of the disc and a movable ball positioned forproviding a variable speed drive between said disc and said outputshaft, first gears connecting said first rack to said disc forrotation'thercof, second gears connecting said first rack to said ballfor moving its position, a second integrator having a second outputshaft, a second movable ball and a second disc, a second rack secured tosaid rail, third gears mounted on said platform and connecting said railto said second disc to rotate said second disc, fourth gears connectingsaid first output shaft to said second ball and a counter actuated bysaid second output shaft, said first and second discs being rotatableabout parallel axes and said first and second output shafts beingrotatable about parallel axes.

9. A calculator comprising a platform mounted for reciprocation along afixed guide rail, a first rack mounted on said platform forreciprocation along an axis perpendicular to said rail, a firstintegrator having a rotating disc, an output shaft rotating at rightangles to the axis of the disc and a movable ball positioned forproviding a variable speed drive between said disc and said outputshaft, first gears connecting said first rack to said ball for movingits position, a second integrator having a second output shaft, a secondmovable ball and a second disc, a second rack secured to said rail,third gears mounted on said platform and connecting said rail to saidsecond disc to rotate said second disc, fourth gears connecting saidfirst output shaft to said second ball, a counter actuated by saidsecond output shaft, and power means for moving said. platform alongsaid rail at a constant rate.

10. A calculator comprising a platform mounted for reciprocation along afixed guide rail, a first rack mounted on said platform forreciprocation along an axis perpendicular to said rail, a firstintegrator having a rotating disc, an output shaft rotating at rightangles to the axis of the disc and a movable ball positioned forproviding a variable speed drive between said disc and said outputshaft, first gears connecting said first rack to said disc for rotationthereof, second gears connecting said first rack to said ball for movingits position, a second integrator having a second output shaft, a secondmovable ball and a second disc, a second rack secured to said rail,third gears mounted on said platform and connecting said rail to saidsecond disc to rotate said second disc, fourth gears connecting saidfirst output shaft to said second ball, a counter actuated by saidsecond output shaft, and auxiliary rotatable manual means forreciprocating said first rack.

References Cited in the file of this patent UNITED STATES PATENTS1,503,824 Fry Aug. 5, 1924 1,875,019 Koeppen Aug. 30, 1932 2,678,772 ImmMay 18, 1954

